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Problem 3.27 Let Q be an operator with a complete set of orthonormal eigenvectors: êlen) = anlen) =anen) (n = 1, 2, 3, ...). (a) Show that can be written in terms of its spectral decomposition: Q = Σan len) (enl. 9n n Hint: An operator is characterized by its action on all possible vectors, so what you must show is that êla) = Σanlen) (enlla), for any vector la). (b) Another way to define a function of f(0) = n n is via the spectral decomposition: f(an) \en) (enl. Show that this is equivalent to Equation 3.100 in the case (3.103) of el. (3.104)

Problem 3.27 Let Q be an operator with a complete set of orthonormal eigenvectors: êlen) = anlen) =anen) (n = 1, 2, 3, ...). (a) Show that can be written in terms of its spectral decomposition: Q = Σan len) (enl. 9n n Hint: An operator is characterized by its action on all possible vectors, so what you must show is that êla) = Σanlen) (enlla), for any vector la). (b) Another way to define a function of f(0) = n n is via the spectral decomposition: f(an) \en) (enl. Show that this is equivalent to Equation 3.100 in the case (3.103) of el. (3.104)

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Problem 3.27 Let Ộ be an operator with a complete set of orthonormal eigenvectors: Qen) = anen) (n = 1, 2, 3, ...). (a) Show that can be written in terms of its spectral decomposition: Ô= Σan len) (enl. n Hint: An operator is characterized by its action on all possible vectors, so what you must show is that = { [ n êla) = for any vector la). (b) Another way to define a function of f(0) = n anlen) (enlla), is via the spectral decomposition: f(an) \en) (enl. (3.103) Show that this is equivalent to Equation 3.100 in the case of e (3.104)
Transcribed Image Text:Problem 3.27 Let Ộ be an operator with a complete set of orthonormal eigenvectors: Qen) = anen) (n = 1, 2, 3, ...). (a) Show that can be written in terms of its spectral decomposition: Ô= Σan len) (enl. n Hint: An operator is characterized by its action on all possible vectors, so what you must show is that = { [ n êla) = for any vector la). (b) Another way to define a function of f(0) = n anlen) (enlla), is via the spectral decomposition: f(an) \en) (enl. (3.103) Show that this is equivalent to Equation 3.100 in the case of e (3.104)
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